two-fund separation in dynamic general ......keywords: portfolio separation, dynamically complete...

TWO-FUND SEPARATION IN DYNAMIC GENERAL EQUILIBRIUM

KARL SCHMEDDERS

KELLOGG SCHOOL OF MANAGEMENT

NORTHWESTERN UNIVERSITY

[email protected]

JANUARY 18, 2005

Abstract. The purpose of this paper is to examine the two-fund separation paradigm

in the context of an infinite-horizon general equilibrium model with dynamically complete

markets and heterogeneous consumers with time and state separable utility functions.

With the exception of the dynamic structure, we maintain the assumptions of the classical

static models that exhibit two-fund separation with a riskless security. In addition to a se-

curity with state-independent payoffs agents can trade a collection of assets with dividends

following a time-homogeneous Markov process. We make no further assumptions about

the distribution of asset dividends, returns, or prices. Agents have equi-cautious HARA

utility functions. If the riskless security in the economy is a consol then agents’ portfolios

exhibit two-fund separation. But if agents can trade only a one-period bond, this result

no longer holds. Examples show this effect to be quantitatively significant. The underly-

ing intuition is that general equilibrium restrictions lead to interest rate fluctuations that

destroy the optimality of two-fund separation in economies with a one-period bond and

result in different equilibrium portfolios.

Keywords: Portfolio separation, dynamically complete markets, consol, one-period bond,

interest rate fluctuation.

I am grateful to Ken Judd, Felix Kubler, Christoph Kuzmics, and Costis Skiadas for helpful discussions.

In addition, I thank Michael Magill and Martine Quinzii for making their unpublished book chapters on

‘Elements of Differential Topology’ available to me.

2 TWO-FUND SEPARATION IN DYNAMIC GENERAL EQUILIBRIUM

1. Introduction

The two-fund separation theorem – among the most remarkable results of classical financetheory – states that investors who must allocate their wealth between a number of riskyassets and a riskless security should all hold the same mutual fund of risky assets. Aninvestor’s risk aversion only affects the proportions of wealth that (s)he invests in the riskymutual fund and the riskless security. But the allocation of wealth across the different riskyassets does not depend on the investor’s preferences.

Canner et al. (1997) point out that popular financial planning advice violates the sepa-ration theorem and call this observation the “asset allocation puzzle.” They document rec-ommendations from different investment advisors who all encourage conservative investorsto hold a higher ratio of bonds to stocks than aggressive investors. Bossaerts et al. (2003)state that the separation result cannot be reconciled with casual empirical observationsand conclude that “most tests of asset pricing models address only the pricing predictions– perhaps because the portfolio choice predictions are obviously wrong.”

These critiques assume that classical two-fund separation, a result from static models suchas the Capital Asset Pricing Model, is applicable to the dynamic nature of modern financialmarkets. Attempting to verify the two-fund separation theorem in actual financial marketsassumes the existence of a riskless asset. In dynamic markets (inflation-indexed) bondswith maturities matching all possible investment horizons would allow investors access tosuch an asset. But bonds with very long maturities and in the limit a consol – a bondyielding safe coupon payments ad infinitum – do not exist. Instead, investors are told tohold cash as a “safe” asset. However, because an investor must continually reinvest cash inthe future at unknown and fluctuating interest rates, cash is safe only in the short term, asemphasized by Campbell and Viceira (2002). Therefore, it is unclear whether one shouldexpect investors’ observed portfolios to satisfy the static two-fund separation property.

In this paper we prove that in a dynamic model of asset trading with a consol the two-fund separation theorem holds. But if only risky assets and a one-period bond (cash)can be traded on financial markets then two-fund separation typically fails. To keep theanalysis as close as possible to the classical static presentations (Cass and Stiglitz (1970)),we use an infinite-horizon general equilibrium model with dynamically complete marketsand heterogeneous consumers with time and state separable utility functions. With theexception of the dynamic structure we maintain the assumptions of the static models. Allagents have HARA utilities with linear absolute risk tolerances having identical slopes. Weassume that there is a security with state-independent payoffs (consol or cash) and that allasset dividends follow a time-homogeneous Markov process, but do not make any furtherassumptions about the distribution of asset dividends, returns, or prices. Efficient equilibriain this model have time-homogeneous consumption and asset price processes. Portfolios areconstant over time. All endogenous variables lie in a finite-dimensional space, so we canapply transversality theory on Euclidean spaces (see Magill and Quinzii (1996b)) to derivegeneric results.

The underlying intuition for the very different portfolio properties is that general equi-librium restrictions create interest rate fluctuation. When the safe asset is a consol these

TWO-FUND SEPARATION IN DYNAMIC GENERAL EQUILIBRIUM 3

fluctuations do not affect portfolios since the agents have a trade-once-and-hold-foreverstrategy. But if the agents can only trade a one-period bond, they have to reestablish theconstant portfolio each period. In that case, interest rate fluctuation destroys the optimalityof two-fund separation and leads to different equilibrium portfolios.

In light of our results, it should come as no surprise that observed investors’ portfolios1

do not satisfy two-fund separation. The aforementioned critiques of Canner et al. (1997)and Bossaerts et al. (2003) are based on the implicit – but incorrect – assumption thatinvestors have access to a truly riskless asset.

The classical papers on two-fund separation are Cass and Stiglitz (1970) and Ross (1978).Both papers address two-fund separation of agents’ portfolio demand. Cass and Stiglitz(1970) provide conditions on agents’ preferences that ensure two-fund separation. Ross(1978) presents conditions on asset return distributions under which two-fund separationholds. Russell (1980) presents a unified approach of Cass and Stiglitz and Ross. Ingersoll(1987) provides a detailed overview of various separation results and highlights the distinc-tion between restrictions on utility functions and restrictions on asset return distributions.Gollier (2001) states the separation result of Cass and Stiglitz in the context of a staticequilibrium model. The first discussion of the two-fund separation idea is Tobin (1958) whoanalyzes portfolio demand in a mean-variance setting. Two-fund separation has been exam-ined in great detail in the CAPM, see for example Black (1972). We can’t possibly do justiceto the huge literature on portfolio separation and mutual fund theorems in the CAPM andjust refer to textbook overviews such as Ingersoll (1987) or Huang and Litzenberger (1988).

Section 2 presents the basic model for our analysis. In Section 3 we show some preliminaryresults. Section 4 develops the two-fund separation theory for our dynamic model, provingthe generalization of the classical static result when the safe asset is a consol and showingthat two-fund separation fails generically when there is only a one-period bond. In Section5 we analyze a variation of the basic model with few assets. Section 6 concludes the analysisand the Appendix contains all technical proofs.

2. The Asset Market Economy

We examine a standard Lucas asset pricing model (Lucas (1978)) with heterogeneousagents and dynamically complete asset markets. Time is indexed by t ∈ N0 ≡ 0, 1, 2, . . ..A time-homogeneous recurrent Markov process of exogenous states (yt)t∈N0 takes values in adiscrete set Y = 1, 2, . . . , S. The Markov transition matrix is denoted by Π. A date-eventσt is the history of shocks up to time t, i.e. σt = (y0, y1, . . . yt). Let Σt denote the possiblehistories σt up to time t and let Σ = ∪tΣt denote all possible histories of the exogenous

1The asset allocation puzzle of Canner et al. (1997) has received a lot of attention in the finance literature.

Among others, Brennan and Xia (2000) claim to solve the puzzle. In a continuous-time portfolio selection

model they examine the behavior of a single investor with a constant relative risk-aversion utility and a

finite horizon who can invest in a single stock, cash, and bonds of different maturities. The interest rate

process is given by an exogenously specified Markov process. They show that the ratio of bonds to stock in

the optimal portfolio is increasing in the risk aversion coefficient. But they neither examine an equilibrium

model nor do they address the classical HARA set-up of Cass and Stiglitz (1970).


states. We denote the predecessor of a date-event σ ∈ Σ by σ∗. The starting node σ0 = y0

has a predecessor σ∗0 = σ−1.There is a finite number of types H = 1, 2, . . . , H of infinitely-lived agents. There is

a single perishable consumption good, which is produced by firms. The agents have noindividual endowment of the consumption good. The firms distribute their output eachperiod to its owners through dividends. Investors trade shares of the firms and othersecurities in order to transfer wealth across time and states. There are J = S assets tradedon financial markets. An asset is characterized by its state-dependent dividends. We denoteasset j’s dividend or payoff by dj : Y → R+, j = 1, . . . , S, which solely depends on thecurrent state y ∈ Y. Each security is either an infinitely-lived (long-lived) asset or a single-period asset. There are J l ≥ 1 long-lived assets in the economy. The remaining S − J l

securities are short-lived assets that are issued in each period. A short-lived asset j issued inperiod t pays dj(y) in period t + 1 if state y occurs and then expires. For ease of expositionwe collect the infinitely lived assets in a set L ≡ 1, . . . , J l and the one-period assets in aset O ≡ J l + 1, . . . , S.

Agent h’s portfolio at date-event σ ∈ Σ is θh(σ) ≡ (θhL(σ), θhO(σ)) = (θh1(σ), . . . , θhS(σ)) ∈RS . His initial endowment asset j prior to time 0 is denoted by θhj

−1, j ∈ L. Each agent haszero initial endowment of the short-lived assets and so these assets are in zero net supply.The infinitely lived assets which represent firm dividends are in unit net supply. Otherfinancial assets, such as a consol, are in zero net supply. We write θL

−1 ≡ (θhL−1)h∈H. The

aggregate endowment of the economy in state y is e(y) =∑

j∈L dj(y). Agent h’s initialendowment of dividends before time 0 is given by ωh(y) =

∑j∈L θhj

−1dj(y) > 0. In order to

avoid unnecessary complications we assume that all agents have nonnegative initial holdingsof each asset and a positive initial holding of at least one asset.

Let q(σ) ≡ (q1(σ), . . . , qS(σ)) be the ex-dividend prices of all assets at date-event σ. Ateach date-event σ = (σ∗y) agent h faces a budget constraint,

ch(σ) =∑

j∈L

θhj(σ∗)(qj(σ) + dj(y)) +∑

j∈O

θhj(σ∗)dj(y)−S∑

j=1

θhj(σ)qj(σ).

Each agent h has a time-separable utility function

Uh(c) = E

∞∑

t=0

βtuh(ct)

,

where c = (c0, c1, c2, . . .) is a consumption process. All agents have the same discount factorβ ∈ (0, 1). We assume that the Bernoulli functions uh : X → R are strictly monotone, twicedifferentiable, and strictly concave on some interval X ⊂ R. Below we discuss conditionsthat ensure equilibrium consumption at every date-event to always lie in the interior of anappropriately chosen consumption set X.

Let the matrix

d =

d1(1) · · · dS(1)...

. . ....

d1(S) · · · dS(S)


represent security dividends or payoffs. The vector of utility functions is U = (U1, . . . , UH).We denote the primitives of the economy by the expression E = (d,X, β, U ; θL

−1, Π).We define a standard notion of a financial market equilibrium.

Definition 1. A financial market equilibrium for an economy E is a process of portfolioholdings (θ1(σ), . . . , θH(σ)) and asset prices (q1(σ), . . . , qJ(σ)) for all σ ∈ Σ satisfyingthe following conditions:

(1)∑H

h=1 θh(σ) =∑H

h=1 θh−1 for all σ ∈ Σ.

(2) For each agent h ∈ H,(θh(σ)

)σ∈Σ

∈ arg maxθ Uh(c) s.t.

ch(σ) =∑

j∈L θhj(σ∗)(qj(σ) + dj(y)) +∑

j∈O θhj(σ∗)dj(y)−∑Sj=1 θhj(σ)qj(σ)

supσ∈Σ |θh(σ)q(σ)| < ∞

3. Equilibrium in Dynamically Complete Markets

We use the Negishi approach (Negishi (1960)) of Judd et al. (2003) to characterize ef-ficient equilibria in our model. Efficient equilibria exhibit time-homogeneous consumptionprocesses and asset prices, that is, consumption allocations and asset prices in date-eventσ = (σ∗y) only depend on the last shock y. We take advantage of this recursivity in ournotation and express the dependence of variables on just the exogenous shock through asubscript. For example, ch

y will denote the consumption of agent h in state y. We de-fine py = u′1(c

1y) to be the price of consumption in state y and p = (py)y∈Y ∈ RS

++ tobe the vector of prices. We denote the S × S identity matrix by IS , Negishi weights byλh, h = 2, . . . , H, and use ⊗ to denote element-wise multiplication of vectors.

If the economy starts in the state y0 ∈ Y at period t = 0, then the Negishi weights andconsumption vectors must satisfy the following equations.

u′1(c1y)− λhu′h(ch

y) = 0, h = 2, . . . , H, y ∈ Y,(1)([IS − βΠ]−1(p⊗ (ch − ωh))

)y0

= 0, h = 2, . . . , H,(2)

H∑

h=1

chy −

H∑

h=1

ωhy = 0, y ∈ Y.(3)

The system of equations (1, 2, 3) has HS+(H−1) unknowns, HS unknown state-contingent,agent-specific consumption levels ch

y , and H − 1 Negishi weights λh. Once we know theconsumption vectors we obtain closed-form expressions for asset prices. The prices of along-lived asset j are given by

(4) qj ⊗ p = [IS − βΠ]−1βΠ(p⊗ dj).

The price of a short-lived asset j in state y is

(5) qjy =

βE

p⊗ djy+|y

py=

βΠy·(p⊗ dj)py

,


where Πy· denotes row y of the matrix Π.

Define the matrix D = (d1, . . . , dJ l, dJ l+1 − qJ l+1, . . . , dS − qS). Judd et al. (2003) show

under conditions ensuring that the matrix D has full rank S and if all transition probabilitiesare strictly positive that, after one initial round of trading at time 0, all agents hold a state-independent portfolio vector Θh ≡ θh

y for all y ∈ Y which is given by

(6) Θh = D−1ch, ∀h ∈ H.

In summary, a financial market equilibrium is a solution to equations (1)–(6).

Judd et al. (2003) impose an Inada condition (limx→0 u′h(x) = ∞) to ensure that thesolutions to equations (1)–(3) yield positive consumption allocations. We cannot make thatassumption here since some of the classical utility functions that yield two-fund separation(e.g., quadratic utility) do not satisfy such an Inada condition. Instead we allow for thepossibility of negative consumption. Those of our utility functions that do not satisfy anInada condition have the property limx→−∞ u′h(x) = ∞. Therefore, equations (1)–(3) havea solution that is bounded below and thus an interior point of a consumption set (interval)X that allows for sufficiently negative consumption. In addition, we need to ensure thatconsumption remains non-satiated since we want to avoid free disposal of income. Wedo not state (tedious) assumptions on fundamentals and refer to Magill and Quinzii (2000,Proposition 3) who show for quadratic utilities how to restrict parameters to ensure positiveand non-satiated consumption. In summary, for an appropriately chosen consumption setX equations (1)–(3) are necessary and sufficient for a consumption allocation of an efficientfinancial market equilibrium. (And ideally we think of specifications of the model thatresult in strictly positive consumption allocations.)

For our analysis of agents’ portfolios we adopt the following two assumptions from Juddet al. (2003).

[A1] All elements of the transition matrix Π are positive,

Π ∈ A ∈ RS×S : Ays > 0 ∀y, s ∈ Y,S∑

s=1

Ays = 1 ∀y ∈ Y.

[A2] Rank[d] = S.

For our application of the parametric transversality theorem (see Appendix A.1) using As-sumption [A1] it is useful to define an open set that is diffeomorphic to the set of admissibletransition matrices,

∆S×(S−1)++ ≡ Ays, y ∈ Y, s ∈ 1, . . . , S − 1 : Ays > 0,

S−1∑

s=1

Ays < 1 ∀y ∈ Y.

We identify transition matrices with elements in ∆S×(S−1)++ . Remark 1 below explains why

it is sensible to have genericity statements with respect to transition probabilities. We wantto examine two-fund separation for the classical families of utility functions and so cannotallow for the popular perturbations of utility functions as, for example, in Cass and Citanna(1998) and Citanna et al. (2004).

The following assumption is not crucial but simplifies our genericity arguments.


[A3] All agents have a positive initial position of the first long-lived asset.

We define the open set ∆H−1++ ≡ x ∈ RH−1

++ :∑H−1

i=1 xi < 1. The assumption requires(θh1−1

)h≥2

∈ ∆H−1++ .

3.1. Some Equilibrium Properties. Economies without aggregate risk are well knownto have equilibria of special structure. For completeness we summarize the equilibriumproperties of such economies.

Proposition 1 (Equilibrium Without Aggregate Risk). Suppose the aggregate endowmentis constant, ey = e for all y ∈ Y.

(1) Consumption allocations and asset prices are the same in every efficient financialmarket equilibrium. Allocations are state-independent. Consumption allocations,asset prices, and portfolios are independent of agents’ utility functions.

(2) If [A1] and [A2] hold, then the equilibrium is unique. Each agent holds constantshares of all long-lived assets (in unit net supply) and does not trade short-livedassets.

Proposition 1 completely characterizes efficient financial market equilibria in economieswithout aggregate uncertainty. Portfolios satisfy what one could call a “one-fund” property.Such a simple equilibrium makes any further analysis of two-fund separation superfluous.Our main results in this paper are for economies with a “riskless” asset. For such economiesthe full-rank assumption [A2] immediately implies that the social endowment in the economyis not constant. That is, there exist y1, y2 ∈ Y such that e(y1) 6= e(y2).

Judd et al. (2003) prove existence of efficient financial market equilibria for generic divi-dends of the short-lived assets. We cannot use this existence result here since the analysisof two-fund separation requires particular dividend structures. Therefore we prove an al-ternative existence result that suits our analysis.

Proposition 2 (Equilibrium with Aggregate Risk). Consider an economy E satisfyingassumption [A2].

(1) If all S assets are long-lived, then the economy E has an efficient financial marketequilibrium.

(2) Suppose also [A1] and [A3] hold. If there are S−1 long-lived assets and a one-periodbond, then E has an efficient equilibrium for generic subsets T ⊂ ∆H−1

++ of initialholdings of the first asset and P ⊂ ∆S×(S−1)

++ of transition matrices.

We prove genericity with respect to transition probabilities as they are a natural choicefor the exogenous parameters in the genericity proofs of our analysis, see Remark 1. Weshow the following lemma also in Appendix A.2.

Lemma 1. In an efficient financial market equilibrium of the economy E the price of aone-period bond, qb, has the following characteristics.

(1) The price qb is constant if and only if the aggregate endowment is constant. In thatcase the bond price equals the discount factor, qb

y = β for all y ∈ Y.


(2) Suppose S ≥ 3 and [A1],[A2] and [A3] hold. For generic subsets T ⊂ ∆H−1++ of

initial holdings of the first asset and P ⊂ ∆S×(S−1)++ of transition matrices the price

of a one-period bond is not a linear function of the aggregate endowment. That is,there do not exist numbers a, f ∈ R such that qb

y = a · ey + f for all y ∈ Y.

At first it may be surprising that Part 2 of the lemma only holds for a generic set oftransition probabilities. We explain why this condition is needed in Remark 1 below.

3.2. Linear Sharing Rules. Linear sharing rules for consumption are the foundation oftwo-fund separation on financial markets. Using standard terminology we say that equilib-rium consumption adheres to a linear sharing rule if it satisfies

chy = mhey + bh ∀h ∈ H, y ∈ Y,

for real numbers mh, bh for all agents h ∈ H. Obviously, in equilibrium it holds that∑Hh=1 mh = 1 and

∑Hh=1 bh = 0. Our results in this paper show that we have to care-

fully distinguish between linear sharing rules with nonzero intercepts and those for whichbh = 0 for all h ∈ H.

Recall that the absolute risk tolerance of agent h’s utility function uh : X → R is definedas Th(c) = −u′h(c)

u′′h(c). Of particular interest for linear sharing rules are utility functions

with linear absolute risk tolerance, that is, Th(c) = ah + ghc, for real numbers gh and ah.

These utility functions comprise the well-known family of HARA (hyberbolic absolute riskaversion) utilities (see Gollier (2001), Hens and Pilgrim (2002)). If all agents have HARAutilities and all their linear absolute risk tolerances have identical slopes, that is, gh ≡ g forall h ∈ H for some slope g, then the agents are said to have equi-cautious HARA utilities.

Utility functions exhibiting linear absolute risk tolerance with constant but nonzero slopefor all agents have the form

[EC] uh(c) =

K(Ah + c

γ

)1−γfor γ 6= 0, 1, c ∈ c ∈ R|Ah + c

γ > 0ln(Ah + c) for γ = 1, c ∈ c ∈ R|Ah + c > 0

with K = sign(1−γγ ) to ensure that u is strictly increasing and strictly concave (on some

appropriate consumption set). The absolute risk tolerance for these utility functions isTh(c) = Ah + c

γ . If γ > 0 and Ah = 0 for all h ∈ H then we have the special case of utilityfunctions with identical constant relative risk aversion (CRRA). If γ = −1 then all agentshave quadratic utility functions.

The limit case for utility functions of the type [EC] as γ →∞ are utility functions withconstant absolute risk aversion (CARA). We write

[CARA] uh(c) = − 1ah

e−ahc

with constant absolute risk tolerance of Th(c) = 1ah ≡ τh.

We need the following lemma for our analysis. It follows from the classical results onPareto-efficient sharing rules by Wilson (1968) and Amershi and Stoeckenius (1983). (SeeGollier (2001) for a textbook treatment of a static equilibrium problem.)


Lemma 2. If all agents have equi-cautious HARA utilities, then the consumption allocationof each agent in an efficient equilibrium satisfies a linear sharing rule.

We calculate the sharing rules directly by solving the Negishi equations (1) for givenweights λh for all h ∈ H. For utility functions of type [EC] equations (1) become

(A1 +

c1

γ

)−γ

− λh

(Ah +

ch

γ

)−γ

= 0, h ∈ H, y ∈ Y,(7)

where we include the trivial equation for agent 1 with weight λ1 = 1 to simplify the subse-quent expressions. Some algebra leads to the following linear sharing rule,

chy = ey ·

((λh)

1γ

∑i∈H(λi)

1γ

)+ γ

(−Ah +

(λh)1γ

∑i∈H(λi)

1γ

∑

i∈HAi

).

Note that for the special case of CRRA utility functions, Ah = 0 for all h ∈ H, the sharingrule has zero intercept. For CARA utility functions the linear sharing rules are as follows.

chy = ey · τh

∑i∈H τ i

+

(τh ln(λh)− τh

∑i∈H τ i

∑

i∈Hτ i ln(λi)

).

Remark 1. Now the necessity of genericity with respect to transition probabilities in Lemma 1is apparent. If transition probabilities are i.i.d. and all agents have HARA utility with γ = 1(but possibly Ah 6= 0), then the linear sharing rule leads to the bond price being a linearfunction of the endowment for any set of dividends and initial portfolios. If ch

y = mhey + bh

for all y ∈ Y then

qy = β

(∑

s∈YΠys

1mhes + bh

)(mhey + bh).

Two-fund separation in models with a one-period bond (see Section 4.2) depends cruciallyon whether the intercept of the sharing rules is zero. We prove the following lemma inAppendix A.2.

Lemma 3. Suppose all agents have equi-cautious HARA utility functions of the type [CARA]or the type [EC] with

∑h∈HAh 6= 0 and [A3] holds. Then, for a generic set T ⊂ ∆H−1

++ ofinitial holdings of the first asset, each agent’s sharing rule is linear with nonzero intercept,that is, bh 6= 0 for all h ∈ H.

4. Two-Fund Separation: Consol vs. One-period Bond

Classical two-fund monetary separation (see, for example, Cass and Stiglitz (1970), In-gersoll (1987), Huang and Litzenberger (1988)) states that investors who must allocate theirwealth between a number of risky assets and a riskless security should all hold the same mu-tual fund of risky assets. An investor’s risk aversion only affects the proportions of wealththat (s)he invests in the risky mutual fund and the riskless security. But the allocation ofwealth across the different risky assets does not depend on the investor’s preferences. In thecontext of our general equilibrium model with several heterogeneous agents this propertystates that the proportions of wealth invested in any two risky assets are the same for allagents in the economy.


Definition 2. Consider an economy E with an asset that has a riskless payoff vector, dSy = 1

for all y ∈ Y. We say that portfolios exhibit two-fund monetary separation if

qjy Θhj

qky Θhk

=qjy Θh′j

qky Θh′k

for all assets j, k 6= S and all agents h, h′ ∈ H in all states y ∈ Y.

All long-lived assets are in unit net supply and so market clearing and the requirementfrom the definition immediately imply that all agents’ portfolios exhibit two-fund separationif and only if Θhj = Θhk for all assets j, k 6= S and all agents h ∈ H. That is, in equilibriumeach agent must have a constant share of every risky asset in the economy. The ratio ofwealth invested in any two risky assets j, k 6= S equals the ratio qj

y/qky of their prices and

thus depends on the state y ∈ Y.

4.1. Infinitely Lived Securities. In this subsection we assume that there are no short-lived assets, that is, J l = S. Then equation (6) immediately yields that the consumptionvector of every agent h is a linear combination of the asset dividends,

(8) ch = (d1, . . . , dS)Θh.

Intuitively, the state-dependent security prices do not affect consumption since the agentsdo not trade the assets. Under the assumptions that all assets are infinitely lived and thatthere is a safe asset we recover the classical two-fund monetary separation result for staticdemands of Cass and Stiglitz (1970) in our dynamic equilibrium context.

Theorem 1 (Two-Fund Separation Theorem). Suppose the economy E has S infinitely livedassets with linearly independent payoff vectors and satisfies Assumptions [A1] and [A2]. Thefirst S − 1 assets are in unit net supply and asset S is a consol in zero net supply. If theagents have equi-cautious HARA utilities then in an efficient equilibrium their portfoliosexhibit two-fund monetary separation.

Proof: Proposition 2 ensures that an efficient equilibrium exists. Lemma 2 implies thatsharing rules are linear and ch

y = mhey + bh ∀h ∈ H, y ∈ Y. Under the assumptions of thetheorem equation (8) has the unique solution ΘhJ = bh and Θhj = mh ∀j = 1, . . . , J − 1. ¤

We can easily extend Theorem 1 to economies with J < S long-lived assets. Marketsare dynamically complete with fewer assets than states and so portfolios exhibit two-fundseparation for J < S when a consol is present.

Kang (2003) observes that the results of Judd et al. (2003) can be generalized to economieswith time-varying positive transition probabilities. In addition, he notices that the resultsalso hold for finite-horizon economies with only long-lived assets. We can use these ob-servations to extend the result of Theorem 1 to economies with a finite time horizon ortime-varying (positive) transition matrices. Either change to our model would affect theNegishi weights λh, h ∈ H, and sharing rules mh, bh, h ∈ H, but two-fund monetaryseparation would continue to hold.


4.2. A One-period Riskless Bond. Now we assume that the riskless asset is not a consolbut instead a one-period bond. In addition the economy has S − 1 infinitely lived assets inunit net supply. In such an economy two-fund monetary separation generically fails evenwhen sharing rules are linear with nonzero intercepts.

Theorem 2. Consider an economy E that satisfies the following conditions.

(i) There are J = S ≥ 3 assets.(ii) There are S − 1 infinitely lived securities in unit net supply. The last asset is a

one-period riskless bond.(iii) Assumptions [A1] – [A3] hold.(iv) All agents have equi-cautious HARA utility functions of the type [CARA] or the type

[EC] with∑

h∈HAh 6= 0.

Then there are generic subsets T ⊂ ∆H−1++ of initial portfolios of the first asset and

P ⊂ ∆S×(S−1)++ of transition matrices such that each agent’s equilibrium portfolio does not

exhibit two-fund monetary separation.

Proof: All agents’ consumption allocations follow a linear sharing rule. Now suppose thatequilibrium portfolios exhibit two-fund monetary separation, so agent h holds a portionϑh ≡ Θhj of all infinitely lived assets j = 1, . . . , S − 1, and ΘhS of the one-period bond.Then equation (6) implies that the portfolio shares must satisfy

(9) mh · e + bh1S = ϑh · e + ΘhS(1S − qS) for all h ∈ H,

where qS denotes the bond price and 1S the vector of all ones. If bh = 0 for all h ∈ H thenΘhS = 0 and mh = ϑh is a solution to this equation. Thus, two-fund monetary separationholds. But Lemma 3 states that under conditions (iii) and (iv) it holds that bh 6= 0 for allh ∈ H for a generic set of initial portfolio holdings.

Now suppose bh 6= 0 for all h. Then any solution to equation (9) must have ΘhS 6= 0.

Thus we can rewrite the equation as

qS =ϑh −mh

ΘhS· e +

ΘhS − bh

ΘhS· 1S .

But now the price of the one-period bond is a linear function of the aggregate endowment.Lemma 1, Part 2, states that for a generic set of initial portfolios and transition matricesthere are no (endogenous) coefficients a, f ∈ R such that qS = a · e + f . Hence, equation(9) does not have a solution generically. The intersection of generic sets is generic. Thestatement of the theorem now follows. ¤

Equation (9) in the proof of the theorem is very instructive in providing intuition forthe lack of two-fund monetary separation when the bond is short-lived. Recall that for aneconomy with a consol the corresponding equation would be as follows.

(10) mh · e + bh1S = ϑh · e + ΘhS1S for all h ∈ H.

So, the only difference that the short-lived bond induces in the portfolio equation is thatthe bond position ΘhS is multiplied by the coupon payment minus the price instead of just


being multiplied by the coupon payment. The economic reason for this difference is thatthe agent does not trade the consol after time 0 but must reestablish the position in theshort-lived bond in every period. This change has no impact on the portfolio weights ifagents’ sharing rules have zero intercept and so the riskless security is not traded. But ifsharing rules have nonzero intercept, then the bond price affects the portfolio weights. Itstill does not destroy two-fund monetary separation if it is a linear function of the socialendowment. But if that relationship does not hold, then the fluctuations of the bond pricelead to a change of the portfolio weights that implement equilibrium consumption.

In summary, fluctuations in the equilibrium interest rates of the short-term bond leadto the breakdown of two-fund monetary separation. These fluctuations affect an agentholding a nonzero bond position in equilibrium because he must rebuild that position inevery period. On the contrary, in an economy with a consol, the agent establishes a positionin the consol at time 0 once and forever. Fluctuations in the price of the consol thereforedo no affect the agent just like he is unaffected by stock price fluctuations. This fact allowshim to hold a portfolio exhibiting two-fund monetary separation.

The fact that interest rate variability has significant economic consequences in a dynamicequilibrium model has also been noted by Magill and Quinzii (2000). They examine aninfinite-horizon CAPM economy with stochastic endowments and observe that with fewerassets than states an Arrow-Debreu allocation can only be achieved if a constant consump-tion stream can be spanned by the payoff matrix. But such a spanning condition maynot hold if the interest rate fluctuates in equilibrium. As a consequence markets will beincomplete.

4.3. A Numerical Example. The purpose of this numerical example is to show thatthe equilibrium interest rate fluctuations in an economy with a one-period bond have aquantitatively nontrivial impact on agents’ portfolios.

Consider an economy with H = 3 agents who have CARA utility functions with coeffi-cients of absolute risk-aversion of 1, 3, and 5, respectively. The agents’ discount factor isβ = 0.95. There are three stocks with the following dividend vectors.

d1 = (1.1, 1.1, 0.9, 0.9),d2 = (1.2, 1.1, 0.9, 0.8),d3 = (0.9, 1.1, 1.1, 0.9).

All elements of the Markov transition matrix are 0.25. The economy starts in state y0 = 1.

The agents’ initial holdings of the three stocks are θhj−1 = 1

3 for h = 1, 2, 3, j = 1, 2, 3.

The state-contingent stock prices are

q1 = (21.64129, 23.09972, 17.79559, 14.63329),

q2 = (21.41381, 22.85691, 17.60854, 14.47947),

q3 = (21.84659, 23.31885, 17.96441, 14.77210).

Suppose that the fourth asset in this economy is a consol. The price vector qc of the consolis then

qc = (22.00141, 23.48411, 18.09172, 14.87679).


The economy satisfies the conditions of Theorem 1 and so agents’ portfolios (written as(Θh1, Θh2,Θh3)) exhibit two-fund monetary separation.

Θ1 = (1523

,1523

,1523

,−0.94401), Θ2 = (523

,523

,523

, 0.34328), Θ3 = (323

,323

,323

, 0.60073).

Suppose the fourth asset is a one-period bond instead of a consol. The bond price qb is

qb = (1.10007, 1.17421, 0.90459, 0.74384).

The agents’ portfolios do not exhibit two-fund monetary separation.

Θ1 = (0.21024, 0.44302, 0.36858,−0.48296)

Θ2 = (0.37809, 0.29345, 0.32052, 0.17562)

Θ3 = (0.41166, 0.26353, 0.31091, 0.30734)

The change in the bond maturity strongly affects portfolios. The least risk-averse agentholds considerably less of the three stocks while the two more risk-averse agents hold con-siderably more of the two stocks than in the economy with a consol.

5. Effectively Complete Markets

Up to this point the analysis has been based on the assumption that the transitionmatrix Π had no zero entries. In the presence of J = S independent assets this assumptionis needed to rule out any possibility of trade in equilibrium after the initial period. Withoutthis assumption there would exist a continuum of portfolio allocations in financial marketssupporting an efficient financial market equilibrium. From an economic point of view thisassumption means that at any time t + 1 any dividend state can occur, no matter whatprevious state in period t occurred. However, it may economically be more intuitive toassume that dividends move more “smoothly” and do not experience jumps of arbitrarysize. For our model this assumption means that all rows of the transition matrix Π containzeros. Any given exogenous state y ∈ Y can only be succeeded by states from a strict subsetof Y. In addition, it may also be more reasonable to assume that markets contain fewerassets than the total number of states but are still complete due to dynamic trading as inKreps (1982). In this section we formalize this idea.

5.1. Economies with J < S Assets. We change two features of our basic model fromSection 2. First, the number J of assets is now smaller than the number S of exogenousstates, J < S. Second, the Markov chain of exogenous states remains recurrent but nowevery row of the transition matrix Π has only exactly J positive entries. The remainingS − J elements in each row are zero. We define the set of successors of a state y ∈ Y as

S(y) = z ∈ Y|Πyz > 0 = Y − z ∈ Y|Πyz = 0.

Our assumption on the transition matrix ensures that the cardinality of all successors setsis identical to J , so |S(y)| = J for all y ∈ Y. We also refer to the set of possible predecessors


of a state y ∈ Y and denote it by P (y) = x ∈ Y|Πxy > 0. For a given set of successor setsS(y), y ∈ Y, we denote the set of permissible transition matrices by

A ∈ RS×S : Ays > 0 ∀s ∈ S(y), y ∈ Y, Ayz = 0 ∀z /∈ S(y), y ∈ Y,∑

s∈S(y)

Ays = 1 ∀y ∈ Y.

We identify such transition matrices again with elements of a set that is diffeomorphic toan open subset of RS×(J−1)

++ . Let S−(y) denote the successor set S(y) without its largestelement and define

∆S×(J−1)++ = Ays, y ∈ Y, s ∈ S−(y) : Ays > 0,

∑

s∈S−(y)

Ays < 1 ∀y ∈ Y.

We denote economies with such restricted transition matrices by Ef .

We can easily adapt our computational approach for the calculation of efficient equilibriato the new model. The system of equations (1, 2, 3) for the computation of the state-contingent equilibrium consumptions does not change. Similarly, the equations (4, 5) forthe asset prices still apply, too. The only change occurs in the agents’ budget constraintswhich are used to determine the portfolio choices. Equations (6) no longer apply but mustbe replaced by the following set of equations,

(11) chx =

∑

j∈L

θhjy (qj

x + djx) +

∑

j∈O

θhjy dj

x − θhxqx ∀x ∈ S(y), y ∈ Y, h ∈ H .

For this set of S × J equations with the S × J unknowns θhjy , y ∈ Y, j = 1, . . . , J , to have a

unique solution the J ×J-payoff matrix for the J states in the set S(y) must have full rankfor all y ∈ Y. The proof of Proposition 2 can be modified to show a generic existence resultunder the assumption that the dividend submatrix dS(y)· has full rank J for all y ∈ Y. Westate the modified assumptions for this model.

[A1’] Each row of Π has exactly J positive elements.[A2’] Rank[d·S(y)] = J for y ∈ Y.

5.2. Consol. We can easily adapt the proof of Theorem 1 in order to establish the corre-sponding result for the revised model.

Theorem 3 (Two-Fund Separation Theorem). Suppose the economy Ef has J infinitelylived assets with linearly independent payoff vectors and satisfies Assumptions [A1’] and[A2’]. The first J − 1 assets are in unit net supply and asset J is a consol in zero netsupply. If the agents have equi-cautious HARA utilities then there is no trade after theinitial period and the agents’ portfolios exhibit two-fund monetary separation.

Proof: The statement of the proposition follows from Lemma 2 and equations (11).Sharing rules are linear, ch

y = mhey + bh ∀h ∈ H, y ∈ Y. Equations (11) simplify tochx =

∑Kj=1 θhj

y (qjx + dj

x) −∑Kj=1 θhj

x qjx ∀x ∈ S(y), since all assets are infinitely lived. Port-

folio shares of θhjy = mh for all j = 1, . . . , K − 1, y ∈ Y and θhK

y = bh for y ∈ Y are theunique solution to the budget equations. ¤

We can easily extend Theorem 3 to economies with K < J long-lived assets. Markets aredynamically complete with fewer than J long-lived assets and so portfolios exhibit two-fundseparation for K < J when a consol is present.


5.3. One-period Bond. The next theorem states that in economies with fewer assets thanstates and a one-period riskless bond there is trade on security markets.

Theorem 4. Consider an economy Ef that satisfies the following conditions.

(i) There are J ≥ 2 assets.(ii) There are J − 1 infinitely lived securities in unit net supply. The last asset is a

one-period riskless bond.(iii) Assumptions [A1’], [A2’], and [A3] hold.(iv) All agents have equi-cautious HARA utility functions of the type [CARA] or the type

[EC] with∑

h∈HAh 6= 0.

Then there are generic subsets T ⊂ ∆H−1++ of initial portfolios of the first asset and Pf ⊂

∆S×(J−1)++ of transition matrices such that the agents trade on security markets.

Without proof we state the failure of two-fund separation when the bond is short-lived.

Corollary 1. Under the assumptions of Theorem 4 portfolios generically do not exhibittwo-fund separation if there are J − 1 ≥ 2 infinitely lived securities in unit net supply.

An intuitive explanation for the occurrence of trade is as follows. Suppose there is notrade in this economy. Then equations (11) simplify to the following equations.

(12) ch −∑

j∈L

Θhjdj −ΘhJ(1S − qJ) = 0 ∀h ∈ H .

If sharing rules have zero intercepts then the bond is not needed for the implementationof the equilibrium consumption allocation and so ΘhJ = 0 for all h ∈ H. There is notrade in the economy and the sharing rules determine the agents’ portfolio positions. Butif sharing rules have nonzero intercepts then any solution of (12) must have ΘhJ 6= 0 andthe consumption vectors ch must lie in the span of the J vectors d1, . . . , dJ−1 and 1S − qJ .

Contrary to the model with S = J and strictly positive transition matrices this propertyis nongeneric when S > J. So equations (12) typically do not have a solution and theremust be trade. The proof of the theorem (see Appendix A.2) formalizes this intuition. Inorder to keep the proof simple we continue to restrict ourselves to the case of equi-cautiousHARA utility functions although the trade result holds for much broader classes of utilityfunctions. (The proof of the corollary is along the lines of our other genericity proofs butthe details are very tedious and thus are omitted.)

5.4. Numerical Example. We modify the example from Section 4.3 to illustrate Theo-rem 4. We eliminate the third stock from the economy and alter the Markov transitionmatrix but do not change any other parameters. The revised transition matrix is

Π =

0.5 0.25 0.25 013

13 0 1

3

0.25 0 0.5 0.250 1

313

13

.


Every state has only three possible successors, J = 3 and S = 4. The stock prices are

q1 = (22.85754, 21.47990, 16.58465, 15.56036),

q2 = (22.88138, 21.41006, 16.53213, 15.46785).

Suppose the third asset in this economy is a consol. The price vector of the consol is

qc = (23.13575, 21.76886, 16.92729, 15.86207).

The economy satisfies the conditions of Theorem 3 and agents’ portfolios (written as(Θh1, Θh2,Θh3)) exhibit two-fund monetary separation,

Θ1 = (1523

,1523

,−0.63461), Θ2 = (523

,523

, 0.23077), Θ3 = (323

,323

, 0.40384).

If markets are completed with a one-period bond then the bond prices are,

qb = (1.05757, 1.05209, 0.89992, 0.84189).

In order to implement the consumption allocation the agents need to engage in some tradewhenever the state of the economy changes. The agents’ portfolios never exhibit two-fundmonetary separation and involve considerable amounts of trade.

y θ11y θ12

y θ13y θ21

y θ22y θ23

y θ31y θ32

y θ33y

1 0.23356 0.46714 −0.83205 0.36961 0.28468 0.30256 0.39682 0.24818 0.529492 0.18441 0.51023 −0.69226 0.38749 0.26901 0.25173 0.42810 0.22076 0.440533 0.52491 0.18352 −0.98206 0.26367 0.38781 0.35711 0.21142 0.42867 0.624944 0.45437 0.24558 −0.83052 0.28932 0.36524 0.30201 0.25631 0.38918 0.52851

6. Conclusion

The celebrated two-fund separation theorem holds in a dynamic general equilibriummodel of asset trading only if agents can trade an infinitely-lived bond. If agents haveaccess to a one-period bond only, two-fund separation typically fails.

On modern financial markets, investors can trade a multitude of financial assets includingmany finite-maturity bonds. But they do not have access to a truly safe asset. Therefore,the critiques of Canner et al. (1997) and Bossaerts et al. (2003), which are based on theimplicit assumption that investors have access to such an asset, are not justified. We cannotexpect two-fund separation to hold when the key assumption is not satisfied.

Our results naturally lead us to question whether modern financial markets may enableinvestors to synthesize a consol through a variety of other assets, thereby leading to two-fund separation. Judd et al. (2004) study this question by examining families of bonds withvariable but finite maturity structures. They argue that for some nongeneric transitionmatrices and dividend structures a finite number of bonds can span a consol. In suchsituations agents hold the same fund of risky stocks. Their computational exercises showthat this result does not hold in general but that such portfolios can be approximatelyoptimal.


Appendix

A.1. Parametric Systems of Equations. We state the theorem on a parametric systemof equations that we use in the genericity proofs below.

Theorem 5. (Parametric Systems of Equations) Let Ω ⊂ Rk, X ⊂ Rn be open setsand let h : Ω ×X → Rm be a smooth function. If n < m and for all (ω, x) ∈ Ω ×X suchthat h(ω, x) = 0 it holds that rank [Dω,xh(ω, x)] = m, then there exists a set Ω∗ ⊂ Ω withΩ−Ω∗ a set of Lebesgue measure zero, such that x ∈ X : h(ω, x) = 0 = ∅ for all ω ∈ Ω∗.

For a detailed discussion of this theorem see Magill and Quinzii (1996a, Paragraph 11;1996b). This theorem is a specialized version of the parametric transversality theorem, seeGuillemin and Pollack (1974, Chapter 2, Paragraph 3) and Mas-Colell (1985, Chapter 8).Billingsley (1986, Section 12) provides a detailed exposition on the k-dimensional Lebesguemeasure in Euclidean space. For an exposition on sets of measure zero see Guillemin andPollack (1974, Chapter 1, Paragraph 7). A set is said to have full measure if its complementis a set of Lebesgue measure zero. An open set of full Lebesgue measure is called generic.

A.2. Proofs. This section contains all proofs that are omitted in the main body of thepaper.

Proof of Proposition 1: Market-clearing and collinearity of marginal utilities imply thatin an efficient equilibrium all agents have state-independent consumption allocations. Definech ≡ ch

y for all y ∈ Y. Equations (2) imply that the agents’ consumption allocations are

ch =([IS − βΠ]−1)y0·ωh

∑s∈Y([IS − βΠ]−1)y0s

for all h ∈ H. The resulting asset prices are for infinitely lived assets, qj = [IS−βΠ]−1 βΠ dj , j ∈L, and for one-period assets, qj = βΠ dj , j ∈ O. If the matrix d has full column rank thenthe solution to equations (6) is unique and gives the agents’ holdings of infinitely-lived assetsj ∈ L in unit net supply,

Θhj =ch

e=

1e

([IS − βΠ]−1)y0·ωh

∑s∈Y([IS − βΠ]−1)y0s

.

(If the matrix d does not have full column rank then this solution is only one in a contin-uum of optimal portfolios.) The agents do not trade any of the other assets. Note that allexpressions in this proof are independent of agents’ utility functions. ¤

Proof of Proposition 2: The existence result of Mas-Colell and Zame (1991) impliesthat there exist equilibrium state-contingent consumption values ch

y , h ∈ H, y ∈ Y, whichsolve the system of equations (1, 2, 3). The critical remaining issue for the existence of anefficient financial market equilibrium is now whether the matrix D has full rank (which isequivalent to the matrix (d1 +q1, . . . , dJ l

+qJ l, dJ l+1, . . . , dS) having full rank). In that case

equations (6) yield the agents’ equilibrium portfolios.If all assets are long-lived then D = d and so D has full rank. We now show that for

economies with a one-period riskless bond the matrix D generically has full rank. If D does


not have full rank then the following set of equations must have a solution.


y) = 0, h = 2, . . . , H, y ∈ Y,(13)[IS − βΠ]−1(p⊗ (ch −

∑

j∈L

θhj−1d

j))

y0

= 0, h = 2, . . . , H,(14)

H∑

h=1

chy −

H∑

h=1

ωhy = 0, y ∈ Y,(15)

qSy py − βΠy· p = 0, y ∈ Y,(16)

∑

j∈L

dj(y)aj + (1− qS(y)) = 0, y ∈ Y.(17)

We denote the system of equations (13)–(17) by F ((ch)h∈H, (λh)h≥2, qS , a; (θh

−1)h≥2, Π·1) =0. The expression F(i) = 0 denotes equations (i). We now show that this system has nosolutions for generic sets of individual asset holdings and transition probabilities.

The system (13)–(17) has HS + (H − 1) + S + (S − 1) endogenous unknowns ch, h ∈ H,λh, h = 2, . . . ,H, qS , and aj , j = 1, . . . , S − 1, in (H − 1)S + (H − 1) + S + S + S

equations. In addition, the function F depends on the (H − 1) + S exogenous parameters(θh1−1

)h≥2

∈ ∆H−1++ and Π·1 where Π−S ∈ ∆S×(S−1)

++ denotes the first S − 1 columns of Π.Assumption [A2] allows us to assume without loss of generality that e1 6= eS .

We now prove that the Jacobian of F taken with respect to ch, qS , θh1−1 and Π·1 has full

row rank (H − 1)S + (H − 1) + S + S + S. Denote by ΛS(x) ∈ RS×S the diagonal matrixwhose diagonal elements are the elements of the vector x ∈ RS . We denote the derivative ofthe budget constraints (14) with respect to the agent’s initial holding in the first infinitelylived asset, − (

[IS − βΠ]−1(p⊗ d1))y0

, by η1. Note that η1 < 0. In order to keep the displaytractable, we show the Jacobian of F for the special case of H = 3.

c1 c2 c3 qS θ21−1 θ31

−1 Π·1F(13)h=2

ΛS

(u′′1(c

1))

ΛS

(−λ2u′′2(c2)

)0 0 0 0 0 S

F(13)h=3ΛS

(u′′1(c

1))

0 ΛS

(−λ3u′′3(c3)

)0 0 0 0 S

F(14)h=20 0 η1 0 1

F(14)h=30 0 0 η1 1

F(15) IS IS IS 0 0 0 0 S

F(16) 0 0 0 0 −βΛ ((p1 − pS) · 1S) S

F(17) 0 0 0 −IS 0 0 0 S

S S S S 1 1 S

The variables above the matrix indicate the variables with respect to which derivativeshave been taken in the column underneath. The numbers to the right and below the matrixindicate the number of rows and columns, respectively. The terms to the left indicate theequations. Missing entries are not needed for the proof.


Now we perform column operations to obtain zero matrices in the first set of columns ofthe Jacobian. The sets of columns for ch, h ∈ H, of the Jacobian then appear as follows.

c1 c2 c3

F(13)h=20 ΛS

(−λ2u′′2(c2)

)0 S

F(13)h=30 0 ΛS

(−λ3u′′3(c3)

)S

F(14)h=20 1

F(14)h=30 1

F(15) IS + Λ(

u′′1 (c1)λ2u′′2 (c2)

)+ Λ

(u′′1 (c1)

λ3u′′3 (c3)

)IS IS S

F(16) 0 0 0 S

The transformed matrix has submatrices of the following ranks.

c1 c2 c3 qS θ21−1 θ31

−1 Π·1F(13)h=2

0 S 0 0 0 0 0 S

F(13)h=30 0 S 0 0 0 0 S

F(14)h=20 0 1 0 1

F(14)h=30 0 0 1 1

F(15) S S S 0 0 0 0 S

F(16) 0 0 0 0 S S

F(17) 0 0 S 0 0 0 S

S S S S 1 1 S

The term DΠ·1F(16) = −βΛ ((p1 − pS) · 1S) has rank S since p1 6= pS due to e1 6= eS .This matrix has full row rank (H − 1)S + (H − 1) + 3S which exceeds the number of

endogenous variables by 1. The function F is defined on open sets with ch ∈ int(X) for allh ∈ H, λh ∈ RS

++ for h ≥ 2, a ∈ RS−1, qS ∈ RS++, (θh1

−1)h≥2 ∈ ∆H−1++ , and Π−S ∈ ∆S×(S−1)

++ .

Hence, F satisfies the hypotheses of the theorem on parametric systems of equations, Theo-rem 5. We conclude that there exist subsets T ⊂ ∆h−1

++ and P ⊂ ∆S×(S−1)++ of full Lebesgue

measure such that the solution set of the system (13)–(17) is empty. The sets T and P areopen. The solutions to (13)–(17) change smoothly with the exogenous parameters. A smallvariation in initial portfolios and probabilities cannot lead to a solvable system if there wasno solution for the original parameters.

We conclude that the matrix D has full rank S and so an equilibrium exists for initialholdings

(θh1−1

)h≥2

∈ T of the first asset and transition matrices such that Π−S ∈ P. ¤

Proof of Lemma 1, Part 1: The price of the one-period bond is qby = β

Πy· u′1(c)u′1(cy)

whereu′1(c) denotes the column vector of utilities u′1(cy), y ∈ Y. If the social endowment e is notconstant, every agent must have nonconstant consumption. Choose y1 ∈ arg minc1

y|y ∈ Ysuch that Πy1s > 0 for some s /∈ arg minc1

y|y ∈ Y. Similarly, choose y2 ∈ arg maxc1y|y ∈

Y such that Πy2s > 0 for some s /∈ arg maxc1y|y ∈ Y. Obviously, y1 6= y2. Then

Πy1· u′1(c) < u′1(cy1) and Πy2· u′1(c) > u′1(cy2) and so qby2

> β > qby1

. If there is no aggregaterisk in the economy, then the bond price equation immediately yields qb = β. ¤


Part 2: The price of the one-period bond in state y ∈ Y satisfies qypy = βΠy·p. If inequilibrium the price is a linear function of the social endowment e, then the following setof equations must have a solution.


y) = 0, h = 2, . . . , H, y ∈ Y,(18)[IS − βΠ]−1(p⊗ (ch −

∑

j∈L

θhj−1d

j))

y0

= 0, h = 2, . . . , H,(19)

H∑

h=1

chy −

H∑

h=1

ωhy = 0, y ∈ Y,(20)

(a ey + f)py − βΠy· p = 0, y ∈ Y.(21)

We denote the system of equations (18)–(21) by F ((ch)h∈H, (λh)h≥2, a, f ; (θh−1)h≥2, Π·1) = 0.

The expression F(i) = 0 denotes equations (i). The system has HS+(H−1)+2 endogenousunknowns ch, λh, h = 2, . . . , H, a, and f and (H − 1)S + (H − 1) + S + S equations. (Notethat the coefficients a and f of the linear price function are endogenous variables.) Inaddition, F depends on the (H − 1) + S exogenous parameters θh1

−1, h = 2, . . . , H, and Π·1.The aggregate endowment is not constant and so we can assume w.l.o.g. that p1 6= pS .

The Jacobian of F taken with respect to ch, θh1−1 and Π·1 is identical to the respective

columns of the corresponding matrix in the proof of Proposition 2. After performing thesame column operations as in that proof we obtain a transformed matrix with submatricesof the following ranks.

c1 c2 c3 θ21−1 θ31

−1 Π·1F(18)h=2

0 S 0 0 0 0 S

F(18)h=30 0 S 0 0 0 S

F(19)h=20 1 0 1

F(19)h=30 0 1 1

F(20) S S S 0 0 0 S

F(21) 0 0 0 0 S S

S S S 1 1 S

This matrix has full row rank (H − 1)S + (H − 1) + S + S which exceeds the number ofendogenous variables by S−2 ≥ 1. The function F is defined on open sets with ch ∈ int(X)for all h ∈ H, λh ∈ RS

++ for h ≥ 2, a, f ∈ R, (θh1−1)h≥2 ∈ ∆H−1

++ and Π−S ∈ ∆S×(S−1)++ . Hence,

F satisfies the hypotheses of Theorem 5 and the proof proceeds as that of Proposition 2. ¤

Proof of Lemma 3: We first consider an economy where all agents have equi-cautiousHARA utility functions of the type [EC]. Then if bh = 0 for some agent h in equilibrium


the following equations must hold.


y) = 0, h = 2, . . . , H, y ∈ Y,(22)([IS − βΠ]−1(p⊗ (ch − ωh))

)y0

= 0, h = 2, . . . , H,(23)

H∑

h=1

chy −

H∑

h=1

ωhy = 0, y ∈ Y,(24)

(−Ah +

(λh)1γ

∑i∈H(λi)

1γ

∑

i∈HAi

)= 0, for one h ∈ H.(25)

We denote the system of equations (22)–(25) by F ((ch)h∈H, (λh)h≥2; (θh−1)h≥2) = 0. The

system has HS + (H − 1) endogenous unknowns ch, h ∈ H and λh, h = 2, . . . , H, and(H − 1)S + (H − 1) + S + 1 equations. In addition, the function F depends on the (H − 1)exogenous parameters θh1

−1, h = 2, . . . , H. We show that the Jacobian of F taken withrespect to ch, λh, and θh1

−1 has full row rank (H − 1)S + (H − 1) + S + 1.

For h ≥ 2 denote the derivative in equation (25) with respect to λh by

η2h =

1γ (λh)

1γ−1 ·

(∑Hi=1(λ

i)1γ − (λh)

1γ

)

(∑Hi=1(λi)

1γ

)2

(H∑

i=1

Ai

).

For h = 1 we cannot take the derivative in (22) with respect to λ1, since it does not appear(it is normalized to one). Instead we differentiate with respect to λ2 and obtain

η21 = −

1γ (λ2)

1γ−1

(∑Hi=1(λi)

1γ

)2

(H∑

i=1

Ai

).

Note that under the condition from the lemma,∑

i∈HAi 6= 0, it holds that η2h6= 0. For the

special case of H = 3 the Jacobian Dch,θh1−1,λhF appears as follows.

c1 c2 c3 θ21−1 θ31

−1 λh

F(22)h=2ΛS

(u′′1(c

1))

ΛS

(−λ2u′′2(c2)

)0 0 0 S

F(22)h=3ΛS

(u′′1(c

1))

0 ΛS

(−λ3u′′3(c3)

)0 0 S

F(23)h=20 η1 0 0 1

F(23)h=30 0 η1 0 1

F(24) IS IS IS 0 0 0 S

F(25) 0 0 0 0 0 η2h

1S S S 1 1 S


After the same column operations as in the proof of Proposition 2 we obtain the followingranks for the various submatrices of the transformed matrix.

c1 c2 c3 θ21−1 θ31

−1 λh

F(18)h=20 S 0 0 0 S

F(18)h=30 0 S 0 0 S

F(19)h=20 1 0 0 1

F(19)h=30 0 1 0 1

F(20) S S S 0 0 0 S

F(21) 0 0 0 0 0 1 1S S S 1 1 S

This matrix has full row rank HS + (H − 1) + 1 which exceeds the number of endogenousvariables by 1. The function F is defined on open sets with ch ∈ int(X) for all h ∈ H,λh ∈ RS

++ for h ≥ 2, and (θh1−1)h≥2 ∈ ∆H−1

++ . Hence, F satisfies the hypotheses of Theorem 5and the proof proceeds as that of Proposition 2. We can perform the proof for each agentand then take the intersection of generic sets which in turn yields a generic set for whichno agent has a linear sharing rule with zero intercept.

In the proof for CARA utilities we must replace equation (25) with the following equation.

(26) τ h ln(λh)− τ h

∑i∈H τ i

∑

i∈Hτ i ln(λi) = 0

for some agent h ∈ H. The proof is now identical to the one for [EC] type utilities. ¤

Proof of Theorem 4: Suppose there is no trade in this economy. Then equations (11)simplify to the following equations,

(27) ch −∑

j∈L

Θhjdj −ΘhJ(1S − qJ) = 0 ∀h ∈ H.

If there is no trade then the following set of equations must have a solution.


y) = 0, h = 2, . . . , H, y ∈ Y,(28)[IS − βΠ]−1(p⊗ (ch −

∑

j∈L

θhj−1d

j))

y0

= 0, h = 2, . . . , H,(29)

H∑

h=1

chy −

H∑

h=1

ωhy = 0, y ∈ Y,(30)

qJy py − βΠy· p = 0, y ∈ Y(31)

c1 −∑

j∈L

Θ1jdj −Θ1J(1S − qJ) = 0.(32)

We denote the system of equations (28)–(32) by F ((ch)h∈H, (λh)h≥2, qJ , Θ1; (θh

−1)h≥2, Π−S) =0. The system has HS+(H−1)+S+J endogenous unknowns ch, h ∈ H, λh, h = 2, . . . , H,qJ , and Θ1j , j = 1, . . . , J, in (H−1)S+(H−1)+S+S+S equations. In addition, F dependson the (H − 1) + S(J − 1) parameters θh1

−1, h = 2, . . . , H, and Πys for y ∈ Y, s ∈ S−(Y ).Define ymin ≡ minS(y) and ymax ≡ maxS(y). The social endowment is not constant across


all the states in S(y), so we can assume w.l.o.g. that pymin 6= pymax . We show that the Ja-cobian of F taken with respect to ch, qJ , θh1

−1 and Π·ymin (the first nonzero element in everyrow of the matrix Π) has full row rank (H − 1)S + (H − 1) + S + S + S. We use ΛS and η1

as in the proof of Proposition 2. Define η2 = −βΛ((pymin − pymax) · 1S

)and note that η2

has full rank S.

c1 c2 c3 qJ θ21−1 θ31

−1 Π·ymin

F(28)h=2ΛS

(u′′1(c

1))

ΛS

(−λ2u′′2(c2)

)0 0 0 0 0 S

F(28)h=3ΛS

(u′′1(c

1))

0 ΛS

(−λ3u′′3(c3)

)0 0 0 0 S

F(29)h=20 0 η1 0 1

F(29)h=30 0 0 η1 1

F(30) IS IS IS 0 0 0 0 S

F(31) 0 0 0 0 η2 S

F(32) IS 0 0 ΛS(Θ1J) 0 0 0 S

S S S S 1 1 S

After the same column operations as in the proof of Proposition 2 we obtain a transformedmatrix having submatrices of the following ranks.

c1 c2 c3 qJ θ21−1 θ31

−1 Π·ymin

F(28)h=20 S 0 0 0 0 0 S

F(28)h=30 0 S 0 0 0 0 S

F(29)h=20 0 1 0 1

F(29)h=30 0 0 1 1

F(30) S S S 0 0 0 0 S

F(31) 0 0 0 0 S S

F(32) S 0 0 S 0 0 0 S

S S S S 1 1 S

The assumptions of Lemma 3 are satisfied and so there exists a generic set T of initial hold-ings of the first asset such that all agents’ sharing rules have nonzero intercepts. Thereforeany solution to equations (32) must satisfy Θ1J 6= 0. Thus, the matrix Λ(Θ1J) has rank S.

The Jacobian has full row rank (H − 1)S + (H − 1) + 3S which exceeds the number ofendogenous variables by S−J > 0. The function F is defined on open sets with ch ∈ int(X)for all h ∈ H, λh ∈ RS

++ for h ≥ 2, Θ1 ∈ RJ , qJ ∈ RS++, (θh1

−1)h≥2 ∈ ∆H−1++ and Π−S−(y) ∈

∆S×(J−1)++ . Hence, F satisfies the hypotheses of Theorem 5 and the proof proceeds as the

previous genericity proofs. ¤

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